Effective permeability conditions for diffusive transport through impermeable membranes with gaps

TL;DR

This work develops a rigorous multiscale framework to quantify how microscale gaps in an impermeable membrane govern macroscale diffusive transport. Using homogenization, it derives explicit effective coupling conditions across the membrane, obtaining a closed-form expression for the effective permeability $P_{\text{eff}}$ in both the long-thin-channel limit and the $\mathcal{O}(1)$ aspect-ratio limit, and reveals memory effects in the time-dependent regime. Numerical validation against full membrane simulations confirms the accuracy of the steady and time-dependent coupling conditions, including early-time Euler–Maclaurin approximations for rapid computation. The results illuminate how membrane microstructure, particularly gap length $L$, spacing $r$, and width $2a$ (through $\delta$ and $\varepsilon$), controls transport through porins in bacterial membranes and provide a general framework applicable to filtration, carbon capture, and other membrane-like systems. Overall, the paper delivers explicit, geometry-dependent macroscopic descriptions from pore-scale structures and highlights regimes where standard constitutive permeability relations remain valid or require memory-bearing corrections.

Abstract

Membranes regulate transport in a wide variety of industrial and biological applications. The microscale geometry of the membrane can significantly affect overall transport through the membrane, but the precise nature of this multiscale coupling is not well characterised in general. Motivated by the application of transport across a bacterial membrane, in this paper we use formal multiscale analysis to derive explicit effective coupling conditions for macroscale transport across a two-dimensional impermeable membrane with periodically spaced gaps, and validate these with numerical simulations. We derive analytic expressions for effective macroscale quantities associated with the membrane, such as the permeability, in terms of the microscale geometry. Our results generalise the classic constitutive membrane coupling conditions to a wider range of membrane geometries and time-varying scenarios. Specifically, we demonstrate that if the exterior concentration varies in time, for membranes with long channels, the transport gains a memory property where the coupling conditions depend on the system history. By applying our effective conditions in the context of small molecule transport through gaps in bacterial membranes called porins, we predict that bacterial membrane permeability is primarily dominated by the thickness of the membrane. Furthermore, we predict how alterations to membrane microstructure, for example via changes to porin expression, might affect overall transport, including when external concentrations vary in time. These results will apply to a broad range of physical applications with similar membrane structures, from medical and industrial filtration to carbon capture.

Figures (14)

• Figure 1: Schematic demonstrating the output of our homogenisation procedure. We systematically derive effective coupling conditions that allow us to accurately model the membrane as a continuous and homogeneous boundary, removing the need to account for each gap individually. Dimensional parameters are defined in § \ref{['s2mathframe']}.
• Figure 2: Full asymptotic structure and nondimensional geometry of the perforated membrane problem in 2D. We show $\Gamma$, the impermeable parts of membrane in red, and its boundary, $\partial \Gamma$ in purple, where we apply no flux conditions. Regions I and V denote our outer regions where $x$ and $y$ are $\mathcal{O}(1)$. Regions II and IV denote the boundary layers on either side of the membrane and regions IIIa, b and c represent our three inner regions. In the $\mathcal{O}(1)$ aspect ratio problem these collapse to one inner region.
• Figure 3: Asymptotic structure of (a) the outer regions, (b) the boundary layer region cell problem and (c) the inner regions of the multiple gaps problem in 2D. In each region we show the asymptotic size of the $(x,y)$ coordinates.
• Figure 4: Schematic to show the effects of the conformal maps $Z_{-}^{-1}$, \ref{['schwarzmap']}, and $\zeta_{-}$, \ref{['zeta1map']}, on the physical domain, with contour lines to show how the space is transformed.
• Figure 5: Cell level Region IV under the mapping \ref{['confmapb2h']} showing the transformation of lines of constant $X$ and $Y_1$.